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Limit theorems for continuous-time random walks with infinite mean waiting times

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Mark M. Meerschaert  and
Hans-Peter Scheffler
Mark M. Meerschaert*
Affiliation:
University of Nevada, Reno
Hans-Peter Scheffler*
Affiliation:
University of Dortmund
*
∗ Postal address: Department of Mathematics, University of Nevada, Reno, NV 89557, USA. Email address: mcubed@unr.edu
∗∗ Postal address: Fachbereich Mathematik, University of Dortmund, 44221 Dortmund, Germany. Email address: hps@math.uni-dortmund.de
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Abstract

A continuous-time random walk is a simple random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Lévy motion subordinated to the hitting time process of a classical stable subordinator. Density functions for the limit process solve a fractional Cauchy problem, the generalization of a fractional partial differential equation for Hamiltonian chaos. We also establish a functional limit theorem for random walks with jumps in the strict generalized domain of attraction of a full operator stable law, which is of some independent interest.

Keywords

Operator self-similar processcontinuous-time random walk

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60K40: Other physical applications of random processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 623 - 638
Copyright
Copyright © Applied Probability Trust 2004 

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